# A Jacobi elliptic integral with cosine

After reduction of a problem, I find myself in front of these integrals, $$\int_0^{2\pi} cn \left( \frac{2K(1/2)}{\pi} \theta, \frac{1}{2} \right) \cos(m \theta) \mathrm{d}\theta,$$ with $$m$$ positive integers. For now,

• I think it vanishes for $$m$$ even,
• I am not sure how to get the value for $$m=1$$,
• I am also not sure whether an expansion of $$\cos(m\theta)$$ then integration by parts would yield something nice (in terms of the case $$m=1$$ hopefully).

I would like to hear your thoughts on these ones!

• $\S 22.11$ of DLMF has several formula for the Fourier series of Jacobi elliptic functions, Apr 26, 2019 at 20:11
• Thank you, I'm looking into it, that might be helpful! Apr 26, 2019 at 20:14
• It worked out, I posted the answer for anyone's interest Apr 26, 2019 at 20:35

As pointed by @achille hui, the Fourier series of Jacobi elliptic functions are known. In particular here, $$cn \left( \frac{2K(1/2)}{\pi} \theta, \frac{1}{2} \right) = \frac{2\sqrt{2}\pi}{K(1/2)} \sum_{n=0}^\infty \frac{e^{-(n+1/2)\pi}}{1+e^{-(2n+1)\pi}} \cos ((2n+1) \theta),$$ in turn, $$\int_0^{2\pi} cn \left( \frac{2K}{\pi} \theta, \frac{1}{2} \right) \cos(m \theta) \mathrm{d}\theta = \begin{cases} \frac{2\sqrt{2}}{K(1/2)} \frac{e^{-m\pi/2}}{1+e^{-m\pi}} \pi^2 &\textrm{if m is odd}\\ 0 &\textrm{otherwise} \end{cases}$$